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Comptes Rendus Mathématique
Volume 337, n° 1
pages 61-66 (juillet 2003)
Doi : 10.1016/S1631-073X(03)00254-1
Received : 7 April 2003 ;  accepted : 6 May 2003
The modular class of a regular Poisson manifold and the Reeb class of its symplectic foliation

Abdelhak  Abouqateb,  Mohamed  Boucetta
Faculté des sciences et techniques, BP 618, Guéliz-Marrkech, Morocco 

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We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. A Riemannian interpretation of the Reeb class will give some geometric criteria which enables one to tell whether the modular class vanishes or not. It also enables one to construct examples of unimodular Poisson manifolds and others which are not unimodular. Finally, we prove that the first leafwise cohomology space is an invariant of Morita equivalence. To cite this article: A. Abouqateb, M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

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Pour une variété de Poisson régulière, il existe une application linéaire naturelle de la 1-cohomologie feuilletée vers la 1-cohomologie de Poisson qui envoie la classe de Reeb du feuilletage symplectique sur la classe modulaire de la structure de Poisson. Nous donnons une interprétation riemannienne de la classe de Reeb ; ce qui permettra d'avoir des critères géométriques pour décider de la nullité ou non de la classe modulaire. Finalement, nous prouvons que la 1-cohomologie feuilletée est un invariant de l'équivalence de Morita. Pour citer cet article : A. Abouqateb, M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 337 (2003).




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